• gruppi algebrici
• line bundles metrizzati
• teorema di rigidità

In this thesis we will be dealing with the arithmetic of abelian varieties,
i.e with the study of the number theory of an abelian variety, or family of those.
In particular, as the title suggests, we will focus on the theory of height functions
and the Mordell-Weil Theorem.

Let us recall that an abelian variety A over a field k is a complete group variety
over k and that a number field is a finite extension of Q.

The Mordell–Weil Theorem states that for an abelian variety A over a number field K, the group
A(K) of K-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group.


The case with A an elliptic curve E and K the rational number field Q is Mordell's theorem,
and it was proved by Louis Mordell in 1922.
Mordell succeeded in establishing the
finiteness of the quotient group E(Q)/2E(Q) which forms a major step in the proof. In fact,
once one has proven that, the process of infinite descent of Fermat allows to conclude easily.

Some years later André Weil took up the subject, producing the generalisation to abelian varieties of any
dimension over arbitrary number fields in his doctoral dissertation published in 1928.
More abstract methods were required to carry out a proof with the same basic structure.

The first chapter of the thesis recalls some fundamental notions and facts on abelian varieties
and abelian schemes following the approach of
J.S. Milne, in chapter V of the book ``Arithmetic Geometry".

Chapter two introduces a few more results on abelian schemes which are less basic. Using them, we give a short proof of
the Weak Mordell-Weil Theorem, i.e. if n is an integer such that all points of A of order
n are rational over K then A(K)/nA(K) is a finite group.

In chapter three we study the theory of height functions.

Height functions have been used as a tool in arithmetic geometry for quite a long time. Their first
appearance dates back to the last third of the nineteenth century in a work by G. Cantor. At the beginning of the twentieth century E. Borel gave the first definition
of heights for ''systeme'' of rational numbers. However, the systematic use of height functions in
arithmetic geometry started only with D.G. Northcott and A. Weil in the late forties.
Since then, Northcott-Weil heights and their generalizations (Arakelov heights and Faltings or
modular heights) have proven to be a key tool for proving major results as
Mordell Conjecture.

Chapter three starts dealing with the classical
theory of height functions: we define the height on projective space and see that it
can be thought as a measure of the arithmetic complexity of points. We then prove a finiteness results and
we show how to generalize the definition of height to a projective variety. When the projective variety is also
an abelian variety, its properties lead to a fundamental fact: the height behaves essentially
quadratically with respect to the group law. This is what we need to deduce the Mordell-Weil Theorem from
the Weak Mordell-Weil Theorem.